The class of quantities which are n-dimensional vectors (first order tensor-quantities). Each vector has an associated spatial dimension and an associated physical dimension. Each vector-quantity can be decomposed into scalar components for a given orthonormal basis of the proper dimension. The physical-dimension of a scalar component of the vector-quantity is equivalent to the physical-dimension of the vector-quantity.
(<=> (Vector-Quantity ?V)
(And (Constant-Quantity ?V)
(Tensor-Quantity ?V)
(= (Tensor-Order ?V) 1)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B)
(Spatial.Dimension ?V))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?V)))
(And (Defined (Vector-Component ?V ?I ?B))
(= (Quantity.Dimension (Vector-Component ?V
?I
?B))
(Quantity.Dimension ?V)))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U)
(Quantity.Dimension ?V)))
(Numeric-Tensor (Magnitude ?V ?U))))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U) (Quantity.Dimension ?V)))
(Numeric-Tensor (Magnitude ?V ?U))))
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B) (Spatial.Dimension ?V))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?V)))
(And (Defined (Vector-Component ?V ?I ?B))
(= (Quantity.Dimension (Vector-Component ?V ?I ?B))
(Quantity.Dimension ?V)))))
(Tensor-Quantity ?V)
(Constant-Quantity ?V)
(<= (Tensor-Order $X 1) (Vector-Quantity $X))
(<=> (Vector-Quantity ?V)
(And (Constant-Quantity ?V)
(Tensor-Quantity ?V)
(= (Tensor-Order ?V) 1)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Basis.Dimension ?B)
(Spatial.Dimension ?V))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?V)))
(And (Defined (Vector-Component ?V ?I ?B))
(= (Quantity.Dimension (Vector-Component ?V
?I
?B))
(Quantity.Dimension ?V)))))
(Forall (?U)
(=> (And (Unit-Of-Measure ?U)
(= (Quantity.Dimension ?U)
(Quantity.Dimension ?V)))
(Numeric-Tensor (Magnitude ?V ?U))))))
(<=> (Unit-Vec ?V)
(And (Vector-Quantity ?V)
(= (Quantity.Dimension ?V) Identity-Dimension)
(= (Dot ?V ?V) 1)))
(Nth-Domain Vector-Component 1 Vector-Quantity)
(=> (= (Vector-Component ?V ?I ?B) ?S) (Vector-Quantity ?V))
(Nth-Domain The-Vector-Quantity 3 Vector-Quantity)
(=> (= (The-Vector-Quantity ?M ?B) ?V) (Vector-Quantity ?V))
(=> (And (Vector-Quantity ?X) (Vector-Quantity ?Y))
(=> (+ ?X ?Y ?Z)
(And (Vector-Quantity ?Z)
(Forall (?B ?I)
(=> (And (Orthonormal-Basis ?B)
(= (Spatial.Dimension ?X)
(Basis.Dimension ?B))
(Positive-Integer ?I)
(=< ?I (Spatial.Dimension ?X)))
(= (Vector-Component ?Z ?I ?B)
(+ (Vector-Component ?X ?I ?B)
(Vector-Component ?Y ?I ?B))))))))
(=> (And (Vector-Quantity ?X) (- ?X ?Y))
(And (Vector-Quantity ?Y)
(= (+ ?X ?Y)
(The-Zero-Vector-Of-Type (Quantity.Dimension ?X)
(Spatial.Dimension ?X)))))
(=> (And (Vector-Quantity ?V1) (Dyad ?T1))
(<=> (Dot ?V1 ?T1 ?V)
(And (Vector-Quantity ?V)
(Forall (?B)
(= ?T
(The-Vector-Quantity (* (Tensor-To-Matrix ?V1
?B)
(Tensor-To-Matrix ?T1
?B))
?B)))
(Forall
(?B ?I ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?T
(Summation
(Lambda
(?I)
(* (Basis.Vec ?B ?I)
(Summation
(Lambda (?J)
(* (Vector-Component
?V1
?J
?B)
(Dyad-Component ?T1
?J
?I
?B)))
1
(Spatial.Dimension ?V1))))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Dyad ?T1) (Vector-Quantity ?V1))
(<=> (Dot ?T1 ?V1 ?V)
(And (Vector-Quantity ?V)
(Forall
(?B)
(= ?T
(The-Vector-Quantity
(Transpose (* (Tensor-To-Matrix ?T1 ?B)
(Transpose (Tensor-To-Matrix
?V1
?B))))
?B)))
(Forall
(?B ?I ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?T
(Summation
(Lambda
(?I)
(* (Basis.Vec ?B ?I)
(Summation
(Lambda (?J)
(* (Dyad-Component ?T1
?I
?J
?B)
(Vector-Component
?V1
?J
?B)))
1
(Spatial.Dimension ?V1))))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Vector-Quantity ?V1) (Vector-Quantity ?V2))
(<=> (Dot ?V1 ?V2 ?S)
(And (Scalar-Quantity ?S)
(Forall (?B)
(= ?S
(Value (* (Tensor-To-Matrix ?V1 ?B)
(Transpose (Tensor-To-Matrix ?V2
?B)))
11)))
(Forall (?B ?J)
(=> (= (Basis.Dimension ?B)
(Spatial.Dimension ?V1))
(= ?S
(Summation (Lambda (?J)
(* (Vector-Component
?V1
?J
?B)
(Vector-Component
?V2
?J
?B)))
1
(Spatial.Dimension ?V1))))))))
(=> (And (Vector-Quantity ?V1) (Scalar-Quantity ?S))
(= (* ?V1 ?S) (* ?S ?V1)))
(=> (And (Scalar-Quantity ?S) (Vector-Quantity ?V1) (* ?S ?V1 ?V))
(And (Vector-Quantity ?V)
(= (Spatial.Dimension ?V1) (Spatial.Dimension ?V))
(Forall (?B ?I)
(=> (= (Basis.Dimension ?B) (Spatial.Dimension ?V1))
(= (Vector-Component ?V ?I ?B)
(* ?S (Vector-Component ?V1 ?I ?B)))))))
(Nth-Domain The-Zero-Vector-Of-Type 3 Vector-Quantity)
(=> (= (The-Zero-Vector-Of-Type ?Spatdim ?Physdim) ?V0)
(Forall (?V)
(=> (Vector-Quantity ?V)
(= (Dot ?V ?V0)
(The-Zero-Scalar-For-Dimension ?Physdim)))))
(<- (Vector-Quantities-Of-Dimensions ?Physim ?Spatdim)
(If (And (Physical-Dimension ?Physim)
(Positive-Integer ?Spatdim))
(Kappa (?Vq)
(And (Vector-Quantity ?Vq)
(= (Spatial.Dimension ?Vq) ?Spatdim)
(= (Quantity.Dimension ?Vq) ?Physim)))))